What Are Reduction Formulas?
In the discipline of integral calculus, a reduction formula is a means of integrating a function, that is, finding the area of a curve under the function or a function that describes the area under that curve. A reduction formula is used when an expression contains an integer parameter that can’t be integrated directly. A reduction formula allows the mathematician to simplify the integral so that it can be evaluated.
What Are The Trig Reduction Formulas?
Trig integral reduction formulas are one type of reduction formula, and they are useful when exponents are too high to work with easily. They can be used with any power of a trig function that is greater than one. The following are the six reduction formulas using the trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant.
- ∫ sinn aXdX = – [(sinn-1 aX cos aX)/an] + (n-1)/n ∫ sinn-2 aXdX
- ∫ cosn aXdX = (cosn-1aX sin aX)/an + (n-1)/n ∫ cosn-2 aXdX
- ∫ tann aXdX = tann-1aX/a(n-1) – ∫ tann-2 aXdX
- ∫ cotn aXdX = – [cotn-1aX/a(n-1)] – ∫ cotn-2 aXdX
- ∫ secn aXdX = (secn-2aX tan aX)/a(n-1) + (n-2)/(n-1) ∫ secn-2 aXdX
- ∫ cscn aXdX = – [(cscn-2aX cot aX)/a(n-1)] + (n-2)/(n-1) ∫ cscn-2 aXdX
When using one of these formulas on an integral with an exponent of 3 or higher, the first result will still include an integral at the end that has to be reduced. If the exponent is greater than one, we simply use the formula again until the final result has an exponent of 1 or no exponent.
Integrating Composite Trig Functions
When one trigonometric function is nested inside another, the expression is called a composite trig function. It can be expressed as F(g(x)). This type of equation can be integrated by substituting u for g(x) when we know how to integrate F and when g(x) differentiates to a constant. Often the product of two or more trig functions can be found using the substitution described below. For example, to solve the equation
F(x) = ∫ sin3X cosX dX
we can substitute the variable u for sinX and du for cosX dX, yielding
F(x) = ∫ u3du
However, if the sine and cosine in the above equation both have exponents, we need to use a different method. For example, if the equation is
F(x) = ∫ sin5x cos3x dx
we can use the identity formula sin2x + cos2x = 1 to change the powers of cosine (except for one) into sinX. This leads to a simpler form of the equation that can be solved with the u and du substitution above.
A different case occurs when both sine and cosine have even exponents. In that case, we need to use the identity formula to change all of the sines to cosines (or vice versa) instead of leaving one. Then we’ll use one of the trigonometric integral reduction formulas above on each part of the equation. Once we do each integral separately, we put them back together in the original equation. With the identity formulas, any trig term can be converted into sines and cosines.